Econ 172A, Fall 2004: Final Examination (I)Instructions.1. The examination has five questions. Answer them all.2. If you do not know how to interpret a question, then ask me.3. You must justify your answers to each question.4. The table below indicates how points will be allocated on the exam.5. Work alone. You may not use notes, books, or calculators.6. You have three hours.7. If you sign the Buckley waiver attached to the exam, then you will b eable to pick up your exam in a public area in Sequoyah 245 when thetests are graded (on or before December 15, 2004). If you do not signthe Buckley waiver, then you will be able to pick up your exam fromthe department undergraduate co ordinator beginning January 18, 2005. Iwill accept requests to reconsider grades only from students who sign theBuckley waiver and submit a request in writing after examining (but notremoving) their examination from Sequoyah Hall. (I try to be especiallycareful when evaluating borderline cases and have changed about five finalgrades in the last 26 years.)Score PossibleI 50II 40III 40IV 60V 60Exam Total 250Course Total 500Grade in Course11. My son Ben is starting a rock band, Lit Fuse. Ben will do the vocalsand play lead guitar. Five of his friends will be in the band. They havedifferent talents. Ben evaluated the relative abilities of his friends andwants to come up with an assignment that maximize s the total quality ofthe band. The table below gives the benefit of assigning a particular boyto a particular instrument. (So, for example, if Adam plays Bass, Alexplays Drums, Chase plays guitar, Isaac plays Keyboard, and Ryan is theRoadie, then the quality if the band is: 10 + 13 + 18 + 12 + 10 = 63.)Bass Drum Guitar Keyboard RoadieAdam 10 12 6 8 5Alex 17 13 10 16 2Chase 15 5 18 11 6Isaac 14 6 16 12 4Ryan 14 6 16 12 10(a) Assume that each boy plays exactly one role (instrument or “roadie”)and each role is assigned to exactly one boy. Find an assignment ofboys to instruments that maximizes the total expected profit. Youranswer should describe which boy should be assigned to which roleand the associated total benefit. You must explain how you arrivedat the answer and why it solves the problem. (Prop e rly using thealgorithm presented in class, with short explanations of the steps,is sufficient. If you do not use the algorithm, you must providedcomplete and detailed arguments to justify your answer.) Pleasenote that the objective is to maximize total benefit.(b) Suppos e that Chase gets a drum set for Christmas and thereforebecomes the band’s drummer. What is the optimal assignment ofroles for the rest of the band? (Again, your answer should describewhich boy should be assigned to which role and the associated totalbenefit. You must explain how you arrived at the answer and whyit solves the problem.)22. A fertilizer company has decided to manufacture a large supply of var-ious plant foods to be sold during the upcoming planting season. Thecompany can invest up to $25,000 in the three basic ingredients: nitrates,which cost $800 per ton; phosphates, which cost $400 per ton; and potash,which costs $1000 per ton. Three standard grades of plant food will beproduced from these ingredients: regular, in which nitrates, phosphates,and potash are combined, respec tively, in a 3:6:1 ratio by weight; extra isa 4:4:1 mixture; super is a 6:4:3 mixture. Regular can be sold for $750 perton. Extra can be sold for $800 per ton. Super can be sold for $900 perton. The company’s objective is to maximize profits (total sales minustotal expenditures for ingredients). Its production c apacity permits it tomanufacture no more than 40 tons of plant food overall.(a) Formulate an LP that would determine how much of each ingredientit should buy and how much of each grade of plant food it shouldproduce.(b) Repeat part (a) subject to the additional condition that the firmcan somehow earn an immediate 10% on all capital not invested innitrates, phosphates, and potash. Hence, the firm can earn $1 oneach $10 not spent on the three ingredients.Both of your answers must include a definition of the variables, in words.The definition must include appropriate units for the variables. You mustalso specify the objective function (and explain why the function you writedown is appropriate) and all constraints (along with a description of howthey correspond to the problem description).33. For what values of A is (x1, x2, x3, x4) = (8, 0, 0, 3) a solution to the linearprogramming problem:max Ax1− 4x2− 6x3+ 5x4subject to x1+ 4x2+ 8x3− 2x4≤ 2−x1+ 2x2+ 4x3+ 3x4≤ 1x ≥ 0You may use any method to answer this question, but you must explainthe method that you use and why it works.44. Jack and Jill live together. Jack works at home and likes to smoke. Jill,who must spend much of the day away from home, hates it when Jacksmokes. Imagine that each day Jack can do one of four things: he canabstain from smoking; he can smoke at 1 PM; he can smoke at 2 PM; orhe can smoke at 3 PM (he never smokes more than once in a day). Jillcan check up on Jack exactly once (at 1, 2, or 3). When Jill checks, shecan detect signs of sm oking in the previous hour (so if Jill checks at 2PM, she’ll know if Jack smoked at 1 or at 2). Assume that Jack and Jillchoose their strategies simultaneously. If Jack do e s not smoke, he receivespayoff 0. If he smokes and Jill does not find out, he receives payoff 1. Ifhe smokes and Jill catches him, he receives payoff -1. The game is zerosum.Answer the questions on the next page. Your answers should providesufficient justification that it is clear you understand the relevant concepts(dominance, value, and so on).5(a) Write down a payoff matrix for this game. (Label the strategies andexplain what they represent.)(b) Does Jack have any dominated strategies? If so, identify the domi-nated strategies and explain why they are dominated.(c) Does Jill have any dominated strategies? If so, identify the dominatedstrategies and explain why they are dominated.(d) Find the pure-strategy security levels of both players.(e) Does the game have an equilibrium in pure strategies? If so, find it.(f) Now assume Jill’s checking strategy is constrained in the followingway: the probability that she checks at 1 PM must be equal to theprobability that she checks at 2 PM. Write down a payoff matrixfor this game . (To do this, assume that Jill has two pure strategies:“check at 3 PM” and “check at 1 PM with probability12and checkat 2 PM with